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Analytification and Tropicalization Over Non-Archimedean Fields

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Analytification and Tropicalization Over Non-Archimedean Fields ANALYTIFICATION AND TROPICALIZATION OVER NON-ARCHIMEDEAN FIELDS ANNETTE WERNER Abstract: In this paper, we provide an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff [BPR11] [BPR13] in the case of curves, we explain a result from [CHW14] comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of most of the results in [GRW14], where a general higher-dimensional theory is developed. In particular, we explain the construction of generalized skeleta in [GRW14] which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the com- parison between polyhedral substructures of Berkovich spaces and tropicalizations. 2010 MSC: 14G22, 14T05 1. Introduction Tropical varieties are polyhedral images of varieties over non-archimedean fields. They are obtained by applying the valuation map to a set of toric coordinates. From the very beginning, analytic geometry was present in the systematic study of tropical varieties, e.g. in [EKL06] where rigid analytic varieties are used. The theory of Berkovich spaces which leads to spaces with nicer topological properties than rigid varieties is even better suited to study tropicalizations. A result of Payne [Pay09] states that the Berkovich space associated to an algebraic variety is homeomorphic to the inverse limit of all tropicalizations in toric varieties. However, individual tropicalizations may fail to capture topological features of an analytic space. arXiv:1506.04846v1 [math.AG] 16 Jun 2015 The present paper is an overview of recent results on the relationship between analytic spaces and tropicalizations. In particular, we address the question of whether a given tropicalization is contained in a Berkovich space as a combinatorial substructure. A novel feature of Berkovich spaces compared to rigid analytic varieties is that they contain interesting piecewise linear combinatorial structures. In fact, Berkovich curves are, very roughly speaking, generalized graphs, where infinite ramifications along a dense set of points is allowed, see [Ber90], chapter 4, [BaRu10] and [BPR13]. 1 2 ANNETTE WERNER In higher dimensions, the structure of Berkovich analytic spaces is more involved, but still they often contain piecewise linear substructures as deformation retracts. This was a crucial tool in Berkovich's proof of local contractibility for smooth analytic spaces, see [Ber99] and [Ber04]. Berkovich constructs these piecewise linear substructures as so-called skeleta of suitable models (or fibration of models). These skeleta basically capture the incidence structure of the irreducible components in the special fiber. In dimension one, Baker, Payne and Rabinoff studied the relationship between tropical- izations and subgraphs of Berkovich curves in [BPR11] and [BPR13]. Their results show that every finite subgraph of a Berkovich curve admits a faithful, i.e. homeomorphic and isometric tropicalization. They also prove that every tropicalization with tropical multiplicity one everywhere is isometric to a subgraph of the Berkovich curve. As a first higher dimensional example, the Grassmannian of planes was studied in [CHW14]. The tropical Grassmannian of planes has an interesting combinatorial structure and is a moduli space for phylogenetic trees. It is shown in [CHW14] that it is homeomorphic to a closed subset of the Berkovich analytic Grassmannian. In [GRW14], the higher dimensional situation is analyzed from a general point of view. This approach is based on a generalized notion of a Berkovich skeleton which is associated to the datum of a semistable model plus a horizontal divisor. This naturally leads to unbounded skeleta and generalizes a well-known construction on curves, see [Tyo12] and [BPR13]. For every rational function f with support in the fixed horizontal divisor it is shown in [GRW14] that the \tropicalization" log jfj factors through a piecewise linear function on the generalized skeleton. This function satisfies a slope formula which is a kind of balancing condition around any 1-codimensional polyhedral face. Moreover, for every generalized skeleton there exists a faithful tropicalization { where faithful in higher di- mension refers to the preservation of the integral affine structures. In dimension one, this can be expressed via metrics. It is also proven that tropicalizations with tropical multiplicity one everywhere admit sections of the tropicalization map, which generalizes the above-mentioned result in [BPR11] on curves. The paper is organized as follows. In section 2 we collect basic facts on Berkovich spaces and tropicalizations, giving references to the literature for proofs and more details. In section 3 we briefly recall some of the results by Baker, Payne and Rabinoff [BPR11] on curves. Section 4 starts with the definition and basic properties of the tropical Grass- mannian. Theorem 4.1 claims the existence of a continuous section to the tropicalization map on the projective tropical Grassmannian. We give a sketch of the proof for the dense torus orbit, where some constructions are easier to explain than in the general case. In section 5 we explain the construction of generalized skeleta from [GRW14], and in section 6 we investigate log jfj for rational functions f. In particular, Theorem 6.5 states the slope formula. Section 7 explains the faithful tropicalization results in higher dimension. ANALYTIFICATION AND TROPICALIZATION OVER NON-ARCHIMEDEAN FIELDS 3 Acknowledgements: The author is very grateful to Walter Gubler for his helpful com- ments. 2. Berkovich spaces and tropicalizations 2.1. Notation and conventions. A non-archimedean field is a field with a non-archime- dean absolute value. Our ground field is a non-archimedean field which is complete with respect to its absolute value. Examples are the field Qp, which is the completion of Q after the p-adic absolute value, finite extensions of Qp and also the p-adic cousin Cp of the complex numbers which is defined as the completion of the algebraic closure of Qp. The field of formal Laurent series k((t)) over an arbitrary base field k is another example. Besides, we can endow any field k with the trivial absolute value (which is one on all non-zero elements). A nice feature of Berkovich's general approach is that it also gives an interesting theory in the case of a trivially valued field. The reason behind this is that Berkovich geometry encompasses also points with values in transcendental field extensions { and those may well carry interesting non-trivial valuations. Let K be a complete non-archimedean field. We write K◦ = fx 2 K : jxj ≤ 1g for the ring of integers in K, and K◦◦ = fx 2 K : jxj < 1g for the valuation ideal. The quotient K~ = K◦=K◦◦ is the residue field of K. The valuation on K× associated to the absolute × value is given by v(x) = − log jxj. By Γ = v(K ) ⊂ R we denote the value group. A variety over K is an irreducible, reduced and separated scheme of finite type over K. 2.2. Berkovich spaces. Let us briefly recall some basic results about Berkovich spaces. This theory was developed in the ground-breaking treatise [Ber90]. The survey papers [Co08], [Du07] and [Tem] provide additional information. For background information on non-archimedean fields and Banach algebras and for an account of rigid analytic geometry see [BGR84] and [Bo14]. For n 2 N und every n−tuple r = (r1; : : : ; rn) of positive real numbers we define the associated Tate algebra as −1 X I I Kfr xg = f aI x : jaI jr ! 0 as jIj ! 1g: n I=(i1;:::;in)2N0 I i in Here we put x = (x1; : : : ; xn), and for I = (i1; : : : ; in) we write x = x 1 : : : x . Moreover, we define jIj = i1 + ::: + in. A Banach algebra A is called K-affinoid, if there exists a surjective K-algebra homo- morphism α : Kfr−1xg ! A for some n and r such that the Banach norm on A is equivalent the quotient seminorm induced by α. If such an epimorphism can be found with r = (1;:::; 1), then A is called strictly K-affinoid. In rigid analytic geometry only strictly K-affinoid algebras are considered (and called affinoid algebras). 4 ANNETTE WERNER The Berkovich spectrum M(A) of an affinoid algebras is defined as the set of all bounded (by the Banach norm), multiplicative seminorms on A. It is endowed with the coarsest topology such that all evaluation maps on functions in A are continuous. If x is a seminorm on an algebra A, and f 2 A, we follow the usual notational convention and write jf(x)j for x(f), i.e. for the real number which we get by evaluating the seminorm x on f. The Shilov boundary of a Berkovich spectrum M(A) of a K-affinoid algebra A is the unique minimal subset Γ (with respect to inclusion) such that for every f 2 A the eval- uation map M(A) ! R≥0 given by x 7! jf(x)j attains its maximum on Γ. Hence for every f 2 A there exists a point z in the Shilov boundary such that jf(z)j ≥ jf(x)j for all x 2 M(A). The Shilov boundary of M(A) exists and is a finite set by [Ber90], Corollary 2.4.5. The Berkovich spectrum of a K-affinoid algebra carries a sheaf of analytic functions which we will not define here. Some care is needed since, very roughly speaking, not all coverings are suitable for glueing analytic functions. Analytic spaces over K are ringed spaces which are locally modelled on Berkovich spectra of affinoid algebras. There is a GAGA-functor, associating to every K-scheme X locally of finite type an analytic space Xan over K.
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